Quantum current algebras associated with rational R-matrix

Slaven Kožić
Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia and School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia

Abstract.

We study quantum current algebra A(¯¯¯¯R) associated with rational R-matrix and we give explicit formulae for the elements of its center at the critical level. Due to Etingof–Kazhdan’s construction, the level c vacuum module Vc(¯¯¯¯R) for the algebra A(¯¯¯¯R) possesses a quantum vertex algebra structure for any complex number c. We prove that any module for the quantum vertex algebra Vc(¯¯¯¯R) is naturally equipped with a structure of restricted A(¯¯¯¯R)-module of level c and vice versa.

Introduction

Let g be a Lie algebra over C equipped with a symmetric invariant bilinear form and let ˆg=g⊗C[t,t−1]⊕CC be the corresponding affine Lie algebra. For any complex number c we associate with ˆg the induced module

Vc(g)=U(ˆg)⊗U(ˆg(⩽0))Cc,whereˆg(⩽0)=∐n⩽0(g⊗t−n)⊕CC

and Cc=C is an U(ˆg(⩽0))-module; the central element C acts on Cc as scalar multiplication by c and g⊗t−n with n⩽0 act trivially. By the results of I. B. Frenkel and Y.-C. Zhu in [6] and B.-H. Lian in [22], the space Vc(g) possesses a vertex algebra structure. Furthermore, any restricted ˆg-module of level c is naturally a module for the vertex algebra Vc(g) and vice versa; see [17, Chapter 6] for more details and references. In this paper, we study a certain quantum version of that result for g=glN.

The notion of quantum vertex operator algebra was introduced by P. Etingof and D. Kazhdan in [4]. They constructed examples of quantum vertex operator algebras by quantizing the quasiclassical structure on Vc(slN) when the classical r-matrix on
slN is rational, trigonometric or elliptic. The corresponding vertex operator map was defined using quantum current T, introduced by N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky in [27], which satisfies a commutation relation of the form111We explain the precise meaning of (1) in Section 2.2.

cf. also Ding’s realization [2] of the quantum affine algebra in type A.
Later on, the theory of quantum vertex algebras was further developed and generalized by H.-S. Li; see [18, 19, 20, 21] and references therein.
Specifically, in [21], an h-adic quantum vertex algebra was constructed on the level c universal vacuum module for a certain cover of the double Yangian ˆDY(sl2) from [13] for any generic c∈C. Moreover, it was proved that any highest weight ˆDY(sl2)-module of level c is naturally equipped with a module structure for that h-adic quantum vertex algebra at the level c.

In this paper, we employ commutation relation (1), where ¯¯¯¯R denotes the
normalized Yang R-matrix, h is a formal parameter and C a central element, to define an associative algebra A(¯¯¯¯R) over C[[h]], which we refer to as quantum current algebra. It is worth noting that the classical limit of (1) coincides with the commutation relation for the affine Lie algebra ˆglN.
We investigate properties of the algebra A(¯¯¯¯R) and, in particular, we use the fusion procedure originated in [11] to give explicit formulae for the elements of its center at the critical level.

Next, we introduce the notion of restrictedA(¯¯¯¯R)-module in parallel with the representation theory of the affine Lie algebras; see, e.g., [17, Chapter 6]. For any complex number c we consider the vacuum moduleVc(¯¯¯¯R) of level c for the algebra A(¯¯¯¯R), which presents an example of restricted A(¯¯¯¯R)-module.
We show that, as a C[[h]]-module, Vc(¯¯¯¯R) is isomorphic to the h-adically completed vacuum module Vc(glN) over the double Yangian for the Lie algebra glN.
Hence, due to aforementioned Etingof–Kazhdan’s construction [4], the C[[h]]-module Vc(¯¯¯¯R)=Vc(glN) possesses a quantum vertex algebra structure.
This structure was recently studied by N. Jing, A. Molev, F. Yang and the author in [9], where the center of Vc(glN) was determined, and also in [14], where a certain connection between quasi Vc(glN)-modules and a class of reflection algebras of A. Molev and E. Ragoucy [25] was established.

The main result of this paper, Theorem 3.9 states that
any module for the quantum vertex algebra Vc(¯¯¯¯R) is naturally equipped with a structure of restricted A(¯¯¯¯R)-module of level c and, conversely, that
any restricted A(¯¯¯¯R)-module of level c is naturally equipped with a structure of module for the quantum vertex algebra Vc(¯¯¯¯R).
Roughly speaking, the proof of the theorem relies on the fact that commutation relation (1) possesses the similar form as the S-locality property, which is one of the fundamental quantum vertex algebra axioms.

We should mention that, starting with the work of E. K. Sklyanin [28], various classes of reflection algebras, which are defined via relations of the form
similar to or same as

R(u−v)B1(u)R(u+v)B2(v)=B2(v)R(u+v)B1(u)R(u−v),

(2)

thus resembling commutation relation (1),
were extensively studied. For more details the reader may consult [7, 10, 15, 16, 25, 23] and references therein. However, in contrast with
(1), reflection equation (2) does not seem to directly give rise to the S-locality property, i.e. to the quantum vertex algebra structure; see [14].

1. Preliminaries

In this section, we recall some properties of the rational R-matrix. Next, we define the (completed) double Yangian for the Lie algebra glN and its vacuum module. Finally, we recall the notions of quantum vertex algebra and module for quantum vertex algebra, which play a central role in this paper.

1.1. Rational R-matrix

Let N⩾2 be an integer and let h be a formal parameter.
We follow [9, Section 2.2] to recall the definition and some basic properties of the rational R-matrix over the ring C[[h]].
Consider the Yang R-matrix over C[[h]],

R(u)=1−hPu−1∈EndCN⊗EndCN[h,u−1],

(1.1)

where 1:x⊗y↦x⊗y is the identity and P:x⊗y↦y⊗x is the permutation operator in CN⊗CN.
There exists a
unique series g(u) in 1+u−1C[[u−1]]
such that

where
ti denotes the transposition applied on the tensor factor i=1,2. As in [9, Section 4.2], we can write (1.5) using the ordered product notation as

(1.6)

where the subscript RL (LR) in (1.6) indicates that the first tensor factor of ¯¯¯¯R12(u)−1 is applied from the right (left) while the second tensor factor of ¯¯¯¯R12(u)−1 is applied from the left (right).222Strictly speaking, notation used in [9] slightly differs. The equalities in (1.6) are expressed therein as

rl¯¯¯¯R12(u)−1¯¯¯¯R12(u+hN)=1andlr¯¯¯¯R12(u)−1¯¯¯¯R12(u+hN)=1.

Indeed, (1.6) is obtained by applying the transposition t1 on the first and t2 on the second equality in (1.5).

1.2. Double Yangian for glN

The double YangianDY(glN) for the Lie algebra glN is the associative algebra over the ring C[[h]] generated by the central element C and the elements
t(±r)ij, where i,j=1,…,N and r=1,2,…, subject to the defining relations

R(u−v)T1(u)T2(v)

=T2(v)T1(u)R(u−v),

(1.7)

R(u−v)T+1(u)T+2(v)

(1.8)

=T+2(v)T1(u)¯¯¯¯R(u−v−hC/2),

(1.9)

see [3, 4, 8, 14].
The elements T(u),T+(u)∈EndCN⊗DY(glN)[[u∓1]] are defined by

T(u)=N∑i,j=1eij⊗tij(u)andT+(u)=N∑i,j=1eij⊗t+ij(u),

where the eij denote the matrix units
and the series tij(u) and t+ij(u) are defined by

tij(u)=δij+h∞∑r=1t(r)iju−randt+ij(u)=δij−h∞∑r=1t(−r)ijur−1.

We indicate a copy of the matrix in the tensor product algebra
(EndCN)⊗m⊗DY(glN) by subscripts,
so that, for example, we have

Tk(u)=N∑i,j=11⊗(k−1)⊗eij⊗1⊗(m−k)⊗tij(u).

(1.10)

In particular, we have m=2 and k=1,2 in defining relations (1.7)–(1.9).

The YangianY(glN) is the subalgebra of DY(glN) generated by the elements t(r)ij, where i,j=1,…,N and r=1,2,… The dual Yangian Y+(glN) is the subalgebra of the double Yangian DY(glN) generated by the elements t(−r)ij, where i,j=1,…,N and r=1,2,…
For any c∈C denote by DYc(glN) the double Yangian at the level c, which is defined as the quotient of the algebra DY(glN) by the ideal generated by the element C−c.

For any integer p⩾1 let Ip(glN) be the left ideal in
DYc(glN) generated by all elements t(r)ij, where i,j=1,…,N and r⩾p. Introduce the completion of the double Yangian DYc(glN) at the level c as the inverse limit

˜DYc(glN)=lim⟵DYc(glN)/Ip(glN).

1.3. Vacuum module over the double Yangian

Let V be an arbitrary C[[h]]-module.
The h-adic topology on V is the topology generated by the basis v+hnV, where v∈V and n∈Z⩾1.
Recall that V is said to be torsion-free if hv≠0 for all nonzero v∈V and that V is said to be separable if ∩m⩾1hmV=0. A C[[h]]-module V is said to be topologically free if it is separable, torsion-free and complete with respect to the h-adic topology. For more details on topologically free C[[h]]-modules see [12, Chapter XVI].

We now introduce the vacuum module over the double Yangian as in [9, Section 4.2].
Let Wc(glN) be the left ideal in
DYc(glN) generated by the elements t(r)ij, where i,j=1,…,N and r=1,2,…
By the Poincaré–Birkhoff–Witt theorem for the double Yangian, see [9, Theorem 2.2], the quotient

DYc(glN)/Wc(glN)

(1.11)

is isomorphic, as a C[[h]]-module, to the dual Yangian Y+(glN).
The vacuum module Vc(glN) at the level c over the double Yangian is
defined as the h-adic completion of quotient (1.11).
The vacuum module Vc(glN) is topologically free ˜DYc(glN)-module. We denote by 1 the image of the unit 1∈DYc(glN) in quotient (1.11).

For positive integers n and m introduce functions depending on the variable z and the
families of variables
u=(u1,…,un) and v=(v1,…,vm) with values in
the space
(EndCN)⊗n⊗(EndCN)⊗m
by

We adopt the following expansion convention. For any variables x1,…,xk expressions of the form (x1+…+xk)s with s<0 should be expanded in nonnegative powers of the variables x2,…,xk.
In particular, expressions of the form (z+ui−vj−n)s with s<0 should be expanded in negative powers of the variable z, so that (1.12) and (1.13) contain only nonnegative powers of the variables u1,…,un and v1,…,vm.
Also, we write

where, due to the aforementioned expansion convention, expressions of the form (ui−vj−n)s with s<0, which appear in (1.14), are expanded in negative powers of the variable ui, so that they contain only nonnegative powers of vj−n.
The functions R12nm(u|v|z) and \reflectbox$→\reflectbox$R$$12nm(u|v|z)
corresponding to Yang R-matrix (1.1) can be defined analogously.

Introduce the operators on (EndCN)⊗n⊗Vc(glN) by

T+[n](u|z)=T+1(z+u1)…T+n(z+un)andT[n](u|z)=T1(z+u1)…Tn(z+un).

Note that, due to our expansion convention, the operator T[n](u|z) contains only nonnegative powers of the variables u1,…,un.
Also, we write

T+[n](u)=T+1(u1)…T+n(un)andT[n](u)=T1(u1)…Tn(un).

(1.15)

Note that both expressions in (1.15) can be viewed as series with coefficients in the double Yangian.
Using defining relations (1.7)–(1.9) one can verify the following equations for the operators on

In relations (1.17)–(1.19), we use superscripts to indicate tensor factors in accordance with
(1.16). For example, T+13[n](u|z1) is applied on tensor factors 1,…,n and n+m+1, and T+23[m](v|z2) is applied on tensor factors n+1,…,n+m and n+m+1. We will often use such notation throughout this paper.

1.4. Quantum vertex algebras

The definitions in this section are presented in the form which we find to be suitable for the setting of this paper. In particular, the next definition of quantum vertex algebra coincides with [9, Definition 3.1]. It presents a minor modification of the original definition of quantum vertex operator algebra given by Etingof and Kazhdan in [4], as explained in [9, Remark 3.2 and 3.4].
For more details on the axiomatics of quantum vertex algebras and related structures the reader may consult [4, 21].
From now on, the tensor products are understood as h-adically completed.

Definition 1.1.

A quantum vertex algebra is a quintuple (V,Y,1,D,S) which satisfies the following axioms:

V is a topologically free C[[h]]-module.

Y is a C[[h]]-module map (the vertex operator map)

Y:V⊗V

→V((z))[[h]]

u⊗v

↦Y(z)(u⊗v)=Y(u,z)v=∑r∈Zurvz−r−1

which satisfies the weak associativity:
for any u,v,w∈V and n∈Z⩾0
there exists r∈Z⩾0
such that

(z0+z2)rY(u,z0+z2)Y(v,z2)w−(z0+z2)rY(Y(u,z0)v,z2)w∈hnV[[z±10,z±12]].

(1.20)

1 is an element of V (the vacuum vector) which satisfies

Y(1,z)v=vfor all v∈V,

(1.21)

and for any v∈V the series Y(v,z)1 is a Taylor series in z with
the property

limz→0Y(v,z)1=v.

(1.22)

D is a C[[h]]-module map V→V
which satisfies

D1=0andddzY(v,z)=[D,Y(v,z)]for all v∈V.

(1.23)

S=S(z) is a C[[h]]-module map
V⊗V→V⊗V⊗C((z)) which satisfies the shift condition

[D⊗1,S(z)]=−ddzS(z),

(1.24)

the Yang–Baxter equation

S12(z1)S13(z1+z2)S23(z2)=S23(z2)S13(z1+z2)S12(z1),

(1.25)

the unitarity condition

S21(z)=S−1(−z),

(1.26)

and the S-locality:
for any u,v∈V and n∈Z⩾0 there exists
r∈Z⩾0 such that

is equivalent to weak associativity (1.20) and S-locality (1.27). In what follows, the (appropriately modified) S-Jacobi identity is used to define the notion of module for a quantum vertex algebra. Originally, modules for h-adic nonlocal vertex algebras, which present a generalization of quantum vertex algebras, as well as modules for some related structures, were introduced and studied by Li; see, e.g., [18, 19, 20, 21] and references therein.

Definition 1.2.

Let (V,Y,1,D,S) be a quantum vertex algebra. A V-module is a pair (W,YW), where W is a topologically free C[[h]]-module
and

YW(z):V⊗W

→W((z))[[h]]

v⊗w

↦YW(z)(v⊗w)=YW(v,z)w=∑r∈Zvrwz−r−1

is a C[[h]]-module map
which satisfies the S-Jacobi identity

z−10δ(z1−z2z0)YW(z1)(1⊗YW(z2))(u⊗v⊗w)

−z−10δ(z2−z1−z0)YW(z2)(1⊗YW(z1))(S(−z0)(v⊗u)⊗w)

=z−12δ(z1−z0z2)YW(Y(u,z0)v,z2)wfor allu,v∈V and w∈W,

(1.29)

and

YW(1,z)w=w for all w∈W.

Let W1 be a topologically free C[[h]]-submodule of W. A pair (W1,YW1) is said to be a V-submodule of W if YW(v,z)w1 belongs to W1 for all v∈V and w1∈W1, where YW1 denotes the restriction and corestriction of YW,

YW1(z)=YW(z)∣∣W1V⊗W1:V⊗W1→W1((z))[[h]].

The next lemma, which we use in the proof of Theorem 3.9, can be viewed as an h-adic analogue of [20, Remark 2.5]; see also [17, Theorem 4.4.5]. Roughly speaking, it is a consequence of [18, Lemma 2.1] and the fact that for any quantum vertex algebra V and V-module W the quotient V/hnV is a weak quantum vertex algebra over C and W/hnW is a V/hnV-module for all n⩾1; see [21] for details.

Lemma 1.3.

Let (V,Y,1,D,S) be a quantum vertex algebra and let W be a topologically free C[[h]]-module equipped with C[[h]]-module map

YW(z):V⊗W

→W((z))[[h]]

v⊗w

↦YW(z)(v⊗w)=YW(v,z)w=∑r∈Zvrwz−r−1

which satisfies YW(1,z)w=w for all w∈W and the weak associativity:
for any u,v∈V, w∈W and n∈Z⩾0
there exists r∈Z⩾0
such that

(z0+z2)rYW(u,z0+z2)YW(v,z2)w

−(z0+z2)rYW(Y(u,z0)v,z2)w∈hnW[[z±10,z±12]].

(1.30)

Then (W,YW) is a V-module.
In particular,
the S-locality holds, i.e.
for any u,v∈V and n∈Z⩾0 there exists
r∈Z⩾0 such that

(z1−z2)rYW(z1)(1⊗YW(z2))(S(z1−z2)(u⊗v)⊗w)

−(z1−z2)rYW(z2)(1⊗YW(z1))(v⊗u⊗w)∈hnW[[z±11,z±12]]for all w∈W.

(1.31)

Proof.
The lemma can be proved by arguing as in [21, Remark 2.16] and using [20, Remark 2.5] and [21, Proposition 2.24].
\qed

2. Quantum current algebras

In this section, we introduce quantum current algebra and study its properties. In particular, we construct an action of the quantum current algebra on the vacuum module over the double Yangian, which is an important ingredient of the proof of our main result, Theorem 3.9 in Section 3. In the end, we give explicit formulae for families of
central elements of the quantum current algebra at the critical level.

2.1. Preliminaries

Our first goal is to introduce the quantum current algebra A(¯¯¯¯R).
However, its defining relations, i.e. the coefficients in commutation relation (2.8) with respect to the variables u and v, are given in terms of certain infinite sums. In order to handle such expressions and employ commutation relation (2.8), we introduce the appropriate completion of the corresponding free algebra.

For any integer N⩾2 let F′(N) be the associative algebra over the ring C[[h]] generated by the elements 1, C, τ(−r)ij and σ(r)ij, where i,j=1,…,N and r=1,2,…, subject to the following defining relations:

C⋅x=x⋅Cand1⋅x=x⋅1=xfor all x∈F′(N).

Hence 1 is the unit and C is a central element in F′(N).
Introduce the elements

τ(r)ij=hrσ(r+1)ij∈F′(N),%
where i,j=1,…,N and r=0,1,…

Let F(N) be the subalgebra of F′(N) generated by the elements 1, C and τ(r)ij, where i,j=1,…,N and r∈Z.
Arrange all τ(r)ij into Laurent series

τij(u)=δij−h∑r∈Zτ(r)iju−r−1∈F(N)[[u±1]],where i,j=1,…,N,

(2.1)

and introduce the elements T(u) in EndCN⊗F(N)[[u±1]] by

T(u)=N∑i,j=1eij⊗τij(u),

(2.2)

where the eij denote the matrix units.

The algebra F′(N) is naturally equipped with the h-adic topology. Let ˜F′(N) be the h-adic completion of F′(N), i.e. ˜F′(N)=F′(N)[[h]]. Note that the induced topology on F(N) from F′(N) does not coincide with the h-adic topology on F(N). For example, the sequence (an)n in F(N) given by an=τ(1)ij+…+τ(n)ij is convergent with respect to the induced topology from F′(N), even though it is not convergent with respect to the h-adic topology on F(N).
Denote by ˜F(N) the completion of F(N) with respect to the induced topology from F′(N). From now on, we consider only the induced topology on the algebra F(N).
Recall the subscript notation from Section 1.2 and the expansion convention from Section 1.3.
Consider the following expressions:

¯¯¯¯¯T[2](u,v)=T1(u)¯¯¯¯R(u−v+hC)−1T2(v)¯¯¯¯R(u−v),

T––[2](u,v)=¯¯¯¯R(−v+u)−1T2(v)¯¯¯¯R(−v+u−hC)T1(u).

Lemma 2.1.

The expressions ¯¯¯¯¯T[2](u,v) and T––[2](u,v)
are well-defined elements of

EndCN⊗EndCN⊗˜F(N)[[u±1,v±1]].

(2.3)

Moreover, for any integer n⩾1 the elements ¯¯¯¯¯T[2](u,v) and T––[2](v,u)333Notice the swapped variables in this term. modulo F(N)∩hnF′(N) belong to

EndCN⊗EndCN⊗F(N)((u))((v)).

(2.4)

Proof.
The matrix entries of
¯¯¯¯R(u−v+hC)−1 and ¯¯¯¯R(u−v)
belong to C[u−1][[hC,h,v]].
For any integer n⩾1 the matrix entries of T(v) modulo F(N)∩hnF′(N) belong to F(N)((v)). Therefore, the expression ¯¯¯¯R(u−v+hC)−1T2(v)¯¯¯¯R(u−v) is a well-defined element of (2.3) and its matrix entries modulo F(N)∩hnF′(N) belong to F(N)[u−1]((v)).
Hence, by left multiplying this expression by T1(u), we conclude that ¯¯¯¯¯T[2](u,v)
is a well-defined element of (2.3) and, furthermore, that ¯¯¯¯¯T[2](u,v) modulo F(N)∩hnF′(N) belongs to (2.4).
The corresponding statements for T––[2](v,u) can be verified analogously.
\qed

By Lemma 2.1, there exist elements
¯¯¯τ(r,s)ijkl and τ––(r,s)ijkl in ˜F